Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
252 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #493 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #2093 |
| prime radical | C22×D8 | Q8 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #12033 | C2 | semidirect |
| 2. #12032 | C2 | semidirect |
| 6. #8827 | C2 | semidirect |
| 7. #7145 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #12033 | 2 |
| #12032 | 2 |
| #11657 | 2 |
| #9663 | 2 |
| #8827 | 2 |
| #7145 | 2 |